Degenerate flag varieties of type A: Frobenius splitting and BW theorem

Let $\Fl^a_\la$ be the PBW degeneration of the flag varieties of type $A_{n-1}$. These varieties are singular and are acted upon with the degenerate Lie group $SL_n^a$. We prove that $\Fl^a_\la$ have rational singularities, are normal and locally complete intersections, and construct a desingularization $R_\la$ of $\Fl^a_\la$. The varieties $R_\la$ can be viewed as towers of successive $\bP^1$-fibrations, thus providing an analogue of the classical Bott-Samelson-Demazure-Hansen desingularization. We prove that the varieties $R_\la$ are Frobenius split. This gives us Frobenius splitting for the degenerate flag varieties and allows to prove the Borel-Weyl type theorem for $\Fl^a_\la$. Using the Atiyah-Bott-Lefschetz formula for $R_\la$, we compute the $q$-characters of the highest weight $\msl_n$-modules.